Properties

Base field \(\Q(\sqrt{19}) \)
Label 2.2.76.1-171.1-e1
Conductor \((3 a)\)
Conductor norm \( 171 \)
CM no
base-change yes: 57.a1,17328.c1
Q-curve yes
Torsion order \( 1 \)
Rank \( 2 \)

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Base field \(\Q(\sqrt{19}) \)

Generator \(a\), with minimal polynomial \( x^{2} - 19 \); class number \(1\).

sage: x = polygen(QQ); K.<a> = NumberField(x^2 - 19)
 
gp: K = nfinit(a^2 - 19);
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-19, 0, 1]);
 

Weierstrass equation

\(y^2+y=x^{3}-x^{2}-2x+2\)
sage: E = EllipticCurve(K, [0, -1, 1, -2, 2])
 
gp: E = ellinit([0, -1, 1, -2, 2],K)
 
magma: E := ChangeRing(EllipticCurve([0, -1, 1, -2, 2]),K);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((3 a)\) = \( \left(-a - 4\right) \cdot \left(-a + 4\right) \cdot \left(a\right) \)
sage: E.conductor()
 
magma: Conductor(E);
 
\(N(\mathfrak{N}) \) = \( 171 \) = \( 3^{2} \cdot 19 \)
sage: E.conductor().norm()
 
magma: Norm(Conductor(E));
 
\(\mathfrak{D}\) = \((171)\) = \( \left(-a - 4\right)^{2} \cdot \left(-a + 4\right)^{2} \cdot \left(a\right)^{2} \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
\(N(\mathfrak{D})\) = \( 29241 \) = \( 3^{4} \cdot 19^{2} \)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
\(j\) = \( -\frac{1404928}{171} \)
sage: E.j_invariant()
 
gp: E.j
 
magma: jInvariant(E);
 
\( \text{End} (E) \) = \(\Z\)   (no Complex Multiplication )
sage: E.has_cm(), E.cm_discriminant()
 
magma: HasComplexMultiplication(E);
 
\( \text{ST} (E) \) = $\mathrm{SU}(2)$

Mordell-Weil group

Rank: \( 2 \)

sage: E.rank()
 
magma: Rank(E);
 

Generators: $\left(-\frac{6}{25} a + \frac{53}{25} : \frac{54}{125} a - \frac{352}{125} : 1\right)$, $\left(a + 3 : -2 a - 10 : 1\right)$

sage: gens = E.gens(); gens
 
magma: gens := [P:P in Generators(E)|Order(P) eq 0]; gens;
 

Heights: 1.67111521371413, 0.464747044349561

sage: [P.height() for P in gens]
 
magma: [Height(P):P in gens];
 

Regulator: 0.0142860183735661

sage: E.regulator_of_points(gens)
 
magma: Regulator(gens);
 

Torsion subgroup

Structure: Trivial
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 

Local data at primes of bad reduction

sage: E.local_data()
 
magma: LocalInformation(E);
 
prime Norm Tamagawa number Kodaira symbol Reduction type Root number ord(\(\mathfrak{N}\)) ord(\(\mathfrak{D}\)) ord\((j)_{-}\)
\( \left(-a - 4\right) \) \(3\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)
\( \left(-a + 4\right) \) \(3\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)
\( \left(a\right) \) \(19\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) .

Isogenies and isogeny class

This curve has no rational isogenies. Its isogeny class 171.1-e consists of this curve only.

Base change

This curve is the base-change of elliptic curves 57.a1, 17328.c1, defined over \(\Q\), so it is also a \(\Q\)-curve.